There are well-known identities involving the Ext bifunctor, coproducts, and products in AB4 abelian categories with enough projectives. Namely, for every such category \[\mathcal{A}\] , given an object X and a set of objects \[{\{ {{\text{A}}_{\text{i}}}\} _{{\text{i}} \in {\text{I}}}}\] , an isomorphism \[Ext_\mathcal{A}^{\text{n}}({ \oplus _{{\text{i}} \in {\text{I}}}}{{\text{A}}_{\text{i}}},{\text{X}}) \cong \prod\nolimits_{{\text{i}} \in {\text{I}}} {Ext_\mathcal{A}^{\text{n}}({{\text{A}}_{\text{i}}},{\text{X}})} \] can be built, where \[Ex{t^{\text{n}}}\] is the nth derived functor of the Hom functor. The goal of this paper is to show a similar isomorphism for the nth Yoneda Ext, which is a functor equivalent to \[Ex{t^{\text{n}}}\] that can be defined in more general contexts. The desired isomorphism is constructed explicitly by using colimits in AB4 abelian categories with not necessarily enough projectives nor injectives, extending a result by Colpi and Fuller in [8]. Furthermore, the isomorphisms constructed are used to characterize AB4 categories. A dual result is also stated.
In [14], a theory of universal extensions in abelian categories is developed, moreover, the notion of Ext-universal object is presented. We show that an Ab3 abelian category which is Ext-small, satisfies the Ab4 condition if, and only if, each object of it is Ext-universal. In particular, this means that there are torsion abelian groups that are not co-Ext-universal in the category of torsion abelian groups. In this sense, we characterize all torsion abelian groups which are co-Ext-universal in such category. Namely, we show that such groups are the ones that admit a decomposition Q ⊕ R, where Q is injective and R is a reduced group on which each p-component is bounded. Contents 1. Introduction 2 2. Preliminaries 3 3. Ab4 vs. Universal extensions 6 4. Ext-universal objects in Ab3 categories 8 5. The category of torsion abelian groups 12 Acknowledgements 25 References 25
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